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f205175 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 | import Mathlib
import OAI.Probability.CriticalPercolation.ProductMeasure.Density
namespace OAI
section
section
open Set MeasureTheory ProbabilityTheory
open Set
open scoped BigOperators
namespace CriticalPercolation.FiniteProduct
open scoped Classical
variable {E : Type*} [Fintype E]
lemma density_modular (w : E → Prop → ℝ) (x y : E → Prop) :
density w x * density w y = density w (x ⊓ y) * density w (x ⊔ y) := by
unfold density
rw [← Finset.prod_mul_distrib, ← Finset.prod_mul_distrib]
apply Finset.prod_congr rfl
intro e _
simp only [Pi.inf_apply, Pi.sup_apply]
by_cases hx : x e <;> by_cases hy : y e <;> simp [hx, hy, mul_comm]
lemma expect_mul_ge (w : E → Prop → ℝ) (hw : ∀ e, ∑ b, w e b = 1)
(hpos : ∀ e b, 0 ≤ w e b) (f g : (E → Prop) → ℝ)
(hf0 : ∀ ω, 0 ≤ f ω) (hg0 : ∀ ω, 0 ≤ g ω)
(hf : Monotone f) (hg : Monotone g) :
expect w f * expect w g ≤ expect w (fun ω => f ω * g ω) := by
have h := fkg f g (density w) (fun ω => density_nonneg hpos ω) hf0 hg0 hf hg
(fun x y => le_of_eq (density_modular w x y))
simpa only [expect, sum_density w hw, one_mul] using h
end CriticalPercolation.FiniteProduct
end
end
end OAI
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